Microsoft Word - 0705LNS151Martin.doc


 

 
  
L I T E R A C Y  &  N U M E R A C Y  S T U D I E S  V O L  1 5  N O  1  2 0 0 6  45 
 

Flexible Mathematical Understanding in an 
Ironworking Apprenticeship Classroom 

 
LYNDON MARTIN, LIONEL LaCROIX and  
LYNDA FOWNES 

 

  
 

Abstract 

The relationship between learning and context has long been an issue 
of interest and concern in the field of adult mathematics education. In 
particular, the questions of whether and how learning can be transferred 
from one context to another, remains a focus of researchers (Lave 1988, 
Lerman 1999, Evans 2000, Carraher and Scheliemann 2002). In this paper 
we look at the mathematical understanding of a group of apprentice 
ironworkers working on a construction task, and explore the flexible nature 
of their understanding. We consider the ways in which they are able to use 
quite formal mathematical ideas and operations and make sense of these 
within the specific context of their trade. Then we discuss how this process is 
more than one of simple transfer. 

Mathematical understanding in workplace training 

In workplace training, mathematical concepts are generally engaged 
with within a ‘task context’ (Wedege 2002:70), where the production of a 
performative solution is required; for example, that of producing a cut 
length of pipe to meet given specifications. To be able to learn and apply 
appropriate mathematics and perform the relevant calculations is something 
of a secondary aim. What matters in the workplace is the resultant product 
of the mathematics. Even in cases where mathematical ideas are introduced 
without the context of a particular task, they are usually framed in terms of 
the specific trade, for example; a worksheet might be titled ‘Mathematics for 
plumbers’ and include mathematical ideas specifically relevant to plumbing. 

This is in contrast to the purpose of mathematical problem-solving as 
posed in the school classroom. Here, the focus is more likely to be the 
development of mathematical concepts, with the problem merely providing 
a context for this to occur (see Wedege 2002). Further, the solution to the 
problem would likely be a calculated answer, rather than an artefact; for 
example, a piece of pipe suitable for the task in hand. As Nunes, Schliemann 
and Carraher (1993:74) note ‘school mathematics is learned mostly as 
written mathematics. Representations of the situation are abandoned as 
much as possible for the sake of generality. In consequence, mathematical 



 F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
  
46 L I T E R A C Y  &  N U M E R A C Y  S T U D I E S   
 

relations represented in school mathematics have poor ties with problem 
situations’. 

Mathematics in the workplace-training situation is generally engaged 
with as an integral part of the practices and demands of a specific trade. As 
a result, what it means to understand and use mathematics becomes more 
complex in this environment. Thus, in workplace training, it is insufficient to 
merely talk about the understanding of a particular mathematical concept 
without considering how such understanding is appropriate to and useable 
for real workplace tasks. In workplace training there are situated forms of 
mathematical understandings at play, which are a consequence of the need 
for mathematics to serve a purpose, to complete a task, or to produce a 
physical output. Pozzi, Noss and Hoyles (1998:117), in their study of the use 
of mathematics by nurses, stated that ‘from our point of view as researchers, 
the question is not: “Do these nurses understand sufficient mathematics?” 
but rather “How do they make sense of the ways a tool represents a patient’s 
state or outputs the correct data?” ’. In a similar way, our question becomes 
‘How does their mathematical understanding enable apprentice ironworkers 
to complete a specific task?’ 

Nunes, Schliemann and Carraher (1993), in considering the 
approaches to a task by apprentice and master carpenters state: 

 
mathematical problem solving involves the use of two types of 
representation (a) representation of the problem situation and (b) 
representation of mathematical relations. A good problem 
solver must be able to connect the two types of representation 
quite easily, pulling out the mathematical relations from a 
problem but also turning the mathematical relations around in 
ways not suggested by the meanings in the problem situation 
(74). 
 
Like Nunes, Schliemann and Carraher (1993), our work recognises 

the vital importance of a workplace apprentice being able to make sense of 
the problem, being able to pull out the mathematical relations, and then 
being able to work with this mathematics with understanding. However, we 
prefer to use the term ‘task’ rather than ‘problem’ as this more closely 
describes the nature of workplace actions. We also introduce a third element 
into the process, that of understanding the specific mathematical practices of 
the particular trade, of knowing how things might actually be done on the 
job, and what this implies for the use of mathematics. 

We are suggesting that mathematical understandings in the 
workplace can take a variety of forms, and that there is a range of ways of 
acting mathematically while working on a task. We continue to acknowledge 
the importance of formal mathematical ideas and operations, what Wedege 



F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
   
M A R T I N ,  L A C R O I X  A N D  F O W N E S   47 
 

(2002) terms ‘school mathematics (what people learn and practice in formal 
education)’ (71), and in this paper we illustrate how a group of ironworkers 
understand and work with such ideas. However, following Noss, Hoyles and 
Pozzi (2000), we suggest that alongside such images, other images or ways of 
acting mathematically, can also be observed and these are vitally important 
for contextualising the formal mathematics. These images are embedded in 
either the nature of the specific problem and its representation or, and of 
particular importance in this paper, in the particular local practices of the 
trade. This third kind of image or representation is something different from 
those offered by Nunes, Schliemann and Carraher (1993), as it involves 
understanding mathematics in a way that makes sense for the task or 
problem in the real life ‘situation context’ (Wedege 2002:64). Understanding 
mathematics in this way can involve a number of elements. These include, 
recognising what is ‘usual’ in the workplace, being familiar with quick ways 
of doing specific calculations, being aware of how calculated answers are 
appropriate to the actual task, and knowing what constraints the carrying 
out of the task might put on the mathematics. For example, in ironwork, 
after calculating the size of the choker needed to lift some assembled beams 
into place, it is common practice to always go up a size – for safety purposes 
and to allow for extra stresses in the lifting of the piece. Thus, not only is 
there the need to be able to calculate the size of choker required, but it is 
also necessary to know that this is actually a minimum specification. In such 
a case the ‘correct’ answer is both the one that is calculated and the one that 
is found by moving up a size from this. However, only one is actually 
useable in completing the task. 

We do not see these three ways of understanding as being alternatives 
or incompatible with one another. Instead we suggest that apprentices are 
likely to engage with these different kinds of understandings many times as 
they progress through their training. In conceptualising understanding, and 
its growth as a multi-faceted and emergent phenomenon, we are influenced 
by the work of Pirie and Kieren (see for example Pirie and Kieren 1994, 
Kieren, Pirie and Gordon-Calvert 1999) and their theory for the dynamical 
growth of mathematical understanding which characterises ‘mathematical 
understanding not as a product of actions but in terms of the actions 
themselves’ (Kieren, Pirie and Gordon-Calvert 1999:212). In particular, and 
of significant relevance to this paper, is the notion that: 

 
growth in understanding is not simply a matter of acting in 
more abstract ways with more and more abstract mathematical 
objects. Such growth in fact entails a dynamic and a connection 
between more and less formal, abstract and sophisticated 
activities. Because such growth in understanding occurs in 
contexts, a study of the growth of understanding must 



 F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
  
48 L I T E R A C Y  &  N U M E R A C Y  S T U D I E S   
 

necessarily take into account the interactions that a person has 
with and in such contexts, including interactions with materials, 
other students and teachers. (Kieren, Pirie and Gordon-Calvert 
1999:229)  

 

In this paper, we focus on one group of apprentice ironworkers and 
explore the ways in which they draw on and work with these three different 
forms of mathematical understandings as they successfully complete a 
challenging construction problem: understandings of the task as posed; 
understandings of the mathematical relations required by the task; and 
understandings of the task as an actual job to carry out. We contend that it 
is the way that these apprentices are able to build an understanding that 
shifts and builds connections across these three dimensions, that leads to 
their success with the task. 

The group and task: planning the assembly of a building 

The larger study, of which this research forms a part, is made up of a 
series of case studies of apprentices training towards qualifications in various 
construction trades in British Columbia, Canada. The case studies involved 
video recorded observations, together with field notes and interviews with 
selected apprentices. Data were collected in the training classroom and 
workshop. Both whole classes and smaller groups of learners were observed, 
depending on the structure of the session. In observing and analysing the 
ways in which the apprentices used their mathematical knowledge in the 
context of workplace tasks we drew on elements of the Pirie-Kieren Theory 
for the dynamic growth of mathematical understanding. 

This paper presents some initial findings and discussion drawn from 
one of these case studies, with a particular focus on identifying the 
mathematical images held, accessed, made, modified and worked with by 
three apprentices Although our conclusions are specific to this case, we 
would suggest that there are implications that may be relevant to the wider 
field. 

The three apprentices discussed in this paper, and known as Joe, 
Andy and Mike, are in a larger class of about 20 students who are in the 
second year of an apprenticeship training program to become credentialed 
ironworkers. The course is part time, and consists of an initial six months at 
an institute of technology in Vancouver, BC, Canada, followed by four 
years in the workplace interspersed with three further six week block release 
sessions back at the institute 

In this session the apprentices have been posed the task of establishing 
the size of a choker sling required to lift an assembled structure of four large 
iron beams into an upright position, and later of determining where the 



F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
   
M A R T I N ,  L A C R O I X  A N D  F O W N E S   49 
 

crane should be positioned to accomplish this. Figure 1 illustrates this as an 
actual event. The structure consists of two upright beams, one top cross-
piece and one middle beam. As can be seen in the photograph, this 
structure is lifted into position using two chokers in a sling arrangement 
around the top beam. It is the size of these chokers that the apprentices have 
been asked to calculate, something that is dependant on the total weight of 
the structure to be lifted. 

Although Figure 1 shows the final result of the task, that is, the 
practical act of the beam being lifted, the session discussed here was 
classroom based and involved the apprentices working with technical plans 
to determine the appropriate configuration prior to its practical 
implementation. At no time in this session were the apprentices involved in 
actually working with or lifting beams. It was a ‘pencil and paper’ exercise, 
but an essential one that would have to be completed on any worksite prior 
to the job being carried out. The apprentices, Joe, Andy and Mike, worked 
closely together for about one hour at a table where they were video and 
audio recorded.  

 

 

Figure 1: The four beams, assembled and being lifted by a 
two-choker sling 

 

The apprentices are required to work with a set of eight technical 
plans, showing different elevations and views of the framework for the 
building. The plans contain all the information necessary to assemble the 
framework, but the form in which this is presented requires considerable 
extraction from, and interpretation of, the various diagrams. Figure 2 is one 
of the plans from which the group often work. It is not necessary to be able 



 F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
  
50 L I T E R A C Y  &  N U M E R A C Y  S T U D I E S   
 

to read the diagram here, it is provided merely to illustrate the complexity of 
such plans. 

 
 

 

Figure 2: An extract from one of the technical plans 

 

The first part of the task requires that the group calculates the total 
weight of the structure to be lifted (that is, the sum of the weights of the four 
beams). Beams of this kind are labelled with two pieces of information: their 
depth and their mass/weight per metre or foot, depending on whether 



F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
   
M A R T I N ,  L A C R O I X  A N D  F O W N E S   51 
 

metric or imperial measures are being used. For example, in this task one of 
the beams ‘W’ is 10 by 21 indicating a depth of 10 inches, and a weight of 
21 pounds per foot. On a plan this would be written simply as ‘W10 x 21’. 
However, what makes the task more complex is the fact that on the set of 
plans, not all the measurements and specifications are in imperial units. 
Some of the eight drawings are solely labelled in metric, whilst others 
actually mix imperial and metric units, depending on what it is that is being 
labelled. This is not an uncommon practice in the workplace, and thus it is 
necessary that apprentices become familiar with using both systems of 
measurement, and are able to move flexibly between the two. For example, 
on one of the diagrams, the specification of a beam is given in metric units – 
W250 x 33. (Note this is not an exact conversion, but the nearest equivalent 
standard available metric sized beam). The question is further complicated 
as the length of the beams (on any of the diagrams) is always given in metric 
units. These lengths are either 3048 millimetres or 9144 millimetres, which 
in this case is simply a conversion from the imperial lengths of 10 and 30 
feet (although this conversion is not printed on any of the plans). Thus, the 
apprentices are working in a mathematically complex environment, 
involving different units and representations that are mixed together in a 
variety of ways.  

Case study 

We join the apprentices at the start of the task, as they are looking at 
a number of the technical plans, locating the relevant beams on these and 
determining their specifications in order to calculate their weights. 

 
Mike: It’s a W 10 by 21. It’s right here (pointing to a drawing of a beam 

on one of the pages of drawings where the specifications are in 
imperial units). 

Joe: It’s not what it says here partner (pointing to the same beam on a 
different plan where metric units are used). 

Mike: Yeah, I know. Maybe that’s f……g metric or something? I don’t know. 
Andy: W 250 by 33? (reading from the same page as Joe) 
 
They begin by working with the drawn elevations, extracting the 

mathematical information contained on them that will be necessary for the 
calculation of the weight of the beam. They understand how to read the 
plans, and they are able to locate the correct beam and its specification from 
a complex diagram like Figure 2. However, Mike and Joe are using two 
different diagrams; one is a plan view (the structure from above), the other a 
cross-sectional elevation of the structure. Because the first of these gives a 
metric specification and the second an imperial, they become concerned 
about what they are working with. Mike has found the beam specification in 



 F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
  
52 L I T E R A C Y  &  N U M E R A C Y  S T U D I E S   
 

imperial units, whilst Joe has it in metric. Although Mike recognises that this 
might be the case, he is not confident, or happy to just assume this is the 
case. The apprentices have an understanding of the visual representation of 
the task, and of how to use the diagrams to locate and extract appropriate 
mathematical information.  
 

Mike: There’s no way that’s 250 inches deep. Its way easier if you ... 
Joe: That’s mills (millimetres). Yeah. 
Mike: Its way easier if we do it with inches. It’s a W 10 by 21. 

 

The comment by Mike ‘There’s no way that’s 250 inches deep’ illustrates 
how he is able to use his understanding of the task as an actual workplace 
job to reflect on the value they have read from the diagram. This is crucial 
for them being able to make sense of what they are doing. It is Mike’s 
knowledge and understanding of how inappropriate a beam with a depth of 
two hundred and fifty inches would be, gained from his experience in the 
field, that instantly alerts him that something is wrong with the number as 
an imperial measure. For Mike the number has a meaning, it is more than 
an abstract object to be operated on – he is drawing on an image of a real 
beam and of what an appropriate depth should be, and is using this to 
inform his mathematical thinking. He now proceeds to try and find the total 
length of the beam from the diagram. 
 

Joe: Okay. See its right there? 
Mike: I’m just trying to find out how long it is. (pause). All I need is one 

measurement. Maybe it’s 3,048 (reading a length from the plan). 
Andy: For what? 

(pause and mumbling from all three apprentices. Joe and Mike 
are looking at the drawing.) 

Andy: That’s what our centimetres will be, 3048. 
Mike: You think so? Oh yeah, it will. It will be, 3,048. We know how long it is. 
Andy: I already got that (laughs). 
Mike: Well. That’s what we were after. 

 

Here they return to the diagrams, and are now looking for the stated 
length of the beam. They locate the correct dimension, 3,048, though they 
think this is centimetres rather than millimetres. As noted earlier, this is a 
metric dimension on the page where the beam specifications are given in 
imperial units. There is a slight sense of them not being sure about this 
measurement, but they are confident that they have the correct information, 
again showing an ability to use the diagrams and to understand what the 



F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
   
M A R T I N ,  L A C R O I X  A N D  F O W N E S   53 
 

numbers on these mean. As noted, 3,048 millimetres is equivalent to ten feet 
in imperial units, and has resulted from the conversion of this to metric 
before adding to the diagram. Again, this illustrates how measurement is a 
messy system in these workplace environments, unlike the structured setting 
of school mathematics. 
 

Joe: So 21 times 348 (working on calculator). You already got this down 
don’t you? (talking to Andy) equals? Is that what you got? (Asking 
Andy). 

Andy: Yeah (he sounds uncertain) 
Joe: In mills? 
Andy: In mills. That’s f…..g 
Joe: Sixty-four (He has actually calculated 3,048 x 21) 
Andy: Sixty-four thousand. So it would be 64. 
Joe: That’s not right. 
Andy: Sixty-four? 
Joe: Kilograms? 

 

Having found the required specifications Joe now begins to calculate 
the weight of the beam. He knows the calculation to perform and obtains 
the correct answer. On seeing that the answer is 64,000, they decide this 
must mean a weight of 64 kilograms, a more likely weight for a beam than 
64,000 kilograms. Again, here it is their knowledge of workplace practice 
and of the actual weight of real beams that informs this, rather than any 
mathematical rule. However, the mistake here is that they have incorrectly 
chosen to multiply a metric length by the pounds per foot specification, 21, 
which has given the wrong answer. 
 

Andy: Kilograms. Yeah. (pause). That can’t be right though. We did something 
wrong. Those beams ain’t a hundred pounds. 

 

Andy is able to bring his understandings of actually working with 
beams into play here  saying ‘That can’t be right though. We did something wrong. 
Those beams ain’t a hundred pounds’. This is a particularly important and 
powerful statement. He is able to not only think practically about the 
question as an actual task, but also is mathematically able to approximately 
convert the calculated weight into an imperial equivalent (albeit a rather 
inaccurate conversion). Without having a working knowledge of beams then 
it would perhaps be unlikely that they would have a sense of the previous 
answer being incorrect. 
 



 F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
  
54 L I T E R A C Y  &  N U M E R A C Y  S T U D I E S   
 

Mike: Well, you have to do it the metric way. You can’t times anything by 40.8. 
(Pause) 

Joe: No, I’m not. I’m just timesing it by the weight per kilogram which is 21. 
Mike: Oh yeah. 
Andy: Is that per metre? 
Mike: No, that’s per foot. 
Andy: Yeah, we’re doing metres though. 
Mike: We got to change something around before we do that. 
Andy: Because if we’re changing that. No… 
Mike: Yeah, yeah yeah. This is per foot. 
 

Although all three apprentices recognise now that the calculation is 
incorrect, they are not instantly able to determine what to do instead. 
However, Mike and Andy use their understandings of the mathematical 
relations involved to realise that they ‘need to change something around’. They 
understand the notions of pounds per foot and kilograms per metre, that 
these have different values and thus some conversion may be necessary. 
There is a sense of reasoning taking place here, of the apprentices 
understanding what they are trying to accomplish, and how to use 
mathematics in this process. They do not simply reach for a formula, nor 
seek help from the instructor, but neither do they use some informal 
method. 
 

Andy: This one that you had over here (indicating the other plan) the 350 
and the 250 times 33, that would be the millimetres. 

Mike: Yeah. 
Joe: Yeah, you’re right. 
Mike: That can’t be 250 by 33. 
Andy: That’s just the… 
Mike: Oh yeah, 33 kilograms per? 

 

Andy realises that they don’t actually need to convert from imperial 
to metric, but instead could simply use the metric specification offered on 
the other diagram, and Mike and Joe agree with this. Mike knows this 
specification is in kilograms, but is not sure ‘per what’. It should be noted 
that no units are printed on the diagram, so the beam is simply labelled as 
W250x33. 
 

Andy: Per? 
Mike: Per metre. (They both nod). Okay, yeah. We’re happening now. Do it 

like this. Times it by 33.  



F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
   
M A R T I N ,  L A C R O I X  A N D  F O W N E S   55 
 

Andy: Thirty-three kilograms 
Mike: Thirty-three kilograms per metre. 

 

Mike is clear about the mathematics to use here. Now that he realises 
they are working with metric measurements, he knows quickly that all that is 
required is to multiply the length by the weight per metre, though it has 
taken him a moment to be sure of this. Interestingly, he even corrects Andy 
who perhaps is not as confident with exactly what the 33 means here. Mike 
is precise and accurate with his statement and again there is a sense of him 
understanding the mathematics he is using, and not merely applying a 
memorised formula. Sarah, a visiting tutor to the session, intervenes here to 
ask how they knew that the specification of the beam was in metric units. 

 

Sarah: How do you know that? 
Mike: Just because if the beam was 250 inches deep it would be a really, really 

big beam and we don’t have any of those. 
 

Mike justifies his thinking by drawing on his knowledge of actual 
beams, of what sizes they come in on the job. He knows that a beam with a 
depth of 250 inches is not realistic. Again, it is this facility to situate the 
mathematics within the real workplace context that allows him to make 
sense of his choice of numbers and subsequent calculation.  

 

Joe: (completing the calculation while Mike talks to Sarah) Three zero 
forty-eight is the length? 

Andy: Yeah. 
(Joe continues to work on the calculator) 

Andy: One hundred point five eight kilograms?  
Joe: Yep. 

 

Joe is able to correctly complete the calculation with an answer of 
100.58 kilograms, and they agree that this is now the weight of the beam in 
kilograms. Although he does not verbalise it here, he has also converted the 
weight into kilograms, as the initial calculation would give the answer in 
grams (due to the use of millimetres rather than metres for the length). 
Although we do not see how he does this, again it seems that he has an 
awareness that a beam with a weight of one hundred kilograms is 
reasonable whereas one with a weight of 1,000 kilograms would not be. 

 



 F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
  
56 L I T E R A C Y  &  N U M E R A C Y  S T U D I E S   
 

Andy: See that makes a little more sense. 
Joe: Multiply two point two. Two hundred and twenty one pounds, that’s more 

like it. 
Andy: That’s a little more like it. 
 

Having got what they are confident is a correct answer, they still do 
not simply accept this, although they seem happy with their use of 
mathematical operations. Instead, they choose to check this answer (which is 
in metric) by converting it to imperial units, as the weight in pounds has a 
greater practical meaning for them. Joe and Andy agree that a weight of 
221 pounds is ‘a little more like it’, thus validating their calculated answer in 
the context of the actual task, and of the size of beams involved. Joe knows 
what a beam should weigh, and that this is an appropriate specification. 
They are thus confident in their use of mathematics, and in the correctness 
of their answer, through drawing on understandings developed from the 
workplace. 

Whilst it is true here that the apprentices could have immediately 
found the weight in pounds by recognising that 3,048 centimetres was the 
same as ten feet, and simply multiplying this by 21, there is nothing in the 
diagrams to indicate this – nor any reason for them to notice this. What is 
striking though, both in the transcript extracts offered here, and in the 
whole session, of which these form a part, is the commitment of the students 
to the completion of the task, and of ensuring their answers are not merely 
mathematically correct, but make sense in the context of actually erecting 
the construction. They are not deterred by the complexity of the diagrams, 
nor that the required information and mathematical operations are not 
immediately obvious. They see problem solving of this kind as a natural part 
of their career choice, and of getting the job done. 

Flexible mathematical understanding in workplace 
training 

Mike, Andy and Joe move continually between three different ways of 
thinking mathematically and demonstrate a flexible understanding of the 
mathematics involved in the task, and of how to work with this in a 
meaningful way. Firstly, they are able to make sense of and work with the 
mathematics of the task, as embedded in the complex set of plans they are 
working from. Secondly, they understand and confidently use the 
mathematical operations required by the task – what to multiply together, 
why, how to convert units and so on. Thirdly, they understand what their 
mathematical answers actually mean in the context of the task, and the 
appropriateness of these for the beam construction they are working with. 



F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
   
M A R T I N ,  L A C R O I X  A N D  F O W N E S   57 
 

What we suggest is important is the way that the three apprentices are 
able to find the solution to the task through bringing these different forms of 
understandings together as they work. This facility to work with 
mathematics in such a fluid and flexible way suggests that they do not see 
more formal mathematical operations as distinct from the context of the 
task, but instead, as embedded within it. Their knowledge of actual beams 
informs their reading of appropriate specifications from the complex plans, 
and also the correctness of their answers. They do not resort to the simple 
use of a memorised formula for the conversion of units, nor do they use 
some informal ad-hoc method that is often the case in the workplace. 
Instead they use standard mathematical operations purposefully, with an 
understanding of what these will achieve and why. When an answer does 
not seem correct, they look to the workplace context to help consider why. 

In observing the ways that Andy, Joe and Mike work, we would 
suggest that what is seen is not a process of transfer – that is ‘a relatively 
passive “carrying over” and deployment of learning from one situation to 
another once learners recognize the “similarity” between those situations’ 
(Carraher and Schliemann 2002:19). Instead we observe what Evans terms 
‘translation’ wherein ‘a making of meaning across discourses’ (2002:223), in 
this case those of mathematics and workplace practice, is occurring. The 
three apprentices are simultaneously able to operate in the domain of 
mathematical operations and also that of the workplace, and to see the task 
they are working on as simultaneously drawing on elements from both sets 
of practices. This flexible understanding is similar to what Carraher and 
Schliemann talk of as an ‘active accommodation of knowledge to the 
demands of the situation’ in which the apprentices have ‘not simply 
unloaded a prior solution from their storehouse of knowledge’ but instead 
have ‘crafted it on the spot, adjusting and adapting their prior knowledge in 
the process’ (Carraher and Schliemann 2002:19). This facility to build or 
craft a powerful and dynamic understanding, that recognises the 
relationship between formal mathematical operations, the representation of 
mathematics in the task posed, and the use of mathematics in the situation 
context, is what enables Andy, Joe and Mike to be successful in the 
completion of the task, and to understand why their answer is appropriate 
and useful. For settings such as construction sites and workplaces, such an 
understanding and awareness is not merely desirable, but essential. As noted 
by Martin, LaCroix and Fownes ‘in the school classroom, an incorrect 
answer will likely result in nothing more than a mark on a piece of paper, 
whereas in the workplace there are real costs associated with such errors’ 
(2005:23). On a building site involving large constructions, such costs may 
be human, as well as financial. For example, the choice of an incorrect size 
of choker could lead to expensive materials being dropped, or to a crane 
tipping over, potentially causing injury or death. 



 F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
  
58 L I T E R A C Y  &  N U M E R A C Y  S T U D I E S   
 

It is beyond the scope of this paper to consider what aspects of the 
learning experiences of Andy, Joe and Mike might have played a significant 
part in the development of such a flexible way of thinking and working. 
However, clearly their training allowed and encouraged them to make 
dynamic connections between what they learned whilst working on actual 
construction sites and what they learned in the trades training classroom. All 
three apprentices are able to make sense of the mathematics they know and 
use, and, perhaps more importantly, they are confident in their 
understanding and see it as sufficient to be able to complete complex tasks of 
the kind discussed here. In the field of adult mathematical learning, where it 
is often the struggles of learners that are reported (eg. Gal 2002, Ingeleton 
and O’Regan 2002) then the example of Andy, Joe and Mike is both 
hopeful and exciting and suggests that adults in apprenticeship training 
programs can be both successful construction workers and successful 
mathematicians.  
 

Acknowledgements 

The research reported in this paper is supported by the Social Science 
and Humanities Research Council of Canada, (SSHRC) through Grants 
#831-2002-0005 and #501-2002-005. We acknowledge the assistance of the 
British Columbia Institute of Technology, the Ironworkers Union Local 97 
and the course instructors. We would also like to thank Andy, Joe and Mike 
for their willingness to be involved in the study. 

 

References 

Carraher, David W and Schliemann, Analucia D (2002) The Transfer 
Dilemma, The Journal of the Learning Sciences, vol 11, no 1, pp 1-24. 

Evans, Jeff (2000) Adults’ Mathematical Thinking and Emotions – A study of 
numerate practices, Routledge Falmer, London. 

Gal, Iddo (2002) Dispositional Aspects of Coping with Interpretive 
Numeracy Tasks, Literacy and Numeracy Studies, vol 11, no 9, pp 47-62. 

Ingleton, Christine and O’Regan, Kerry (2002) Recounting Mathematical 
Experiences: Emotions in mathematics learning, Literacy and Numeracy 
Studies, vol 11, no 9, pp 95-108. 

Kieren, Thomas E, Pirie, Susan E.B and Gordon-Calvert, Lynn (1999) 
Growing Minds, Growing Mathematical Understanding: 
Mathematical understanding, abstraction and interaction, in Burton, 
Leone, ed, Learning Mathematics: From hierarchies to networks, Falmer 
Press, London, pp 209-231. 

Lave, Jean (1988) Cognition in Practice, Cambridge University Press, 
Cambridge. 



F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
   
M A R T I N ,  L A C R O I X  A N D  F O W N E S   59 
 

Lerman, Stephen (1999) Culturally Situated Knowledge, in Burton, Leone, 
ed, Learning Mathematics: From Hierarchies to Networks, Falmer Press, 
London, pp 93-107. 

Martin, Lyndon C, LaCroix, Lionel and Fownes, Lynda (2005) Folding 
Back and the Growth of Mathematical Understanding in Workplace 
Training, Adults Learning Mathematics, vol 1, no 1, pp 19-35. 

Noss, Richard, Hoyles, Celia, and Pozzi, Stefano (2000) Working 
Knowledge: Mathematics in Use, in Bessot, Annie and Ridgway, 
David, eds, Education for Mathematics in the Workplace, Kluwer Academic 
Publishers, Dordrecht, Netherlands, pp 17-36. 

Nunes, Terezinha, Schliemann, Analucia D, and Carraher, David W (1993) 
Street Mathematics and School Mathematics, Cambridge University Press, 
Cambridge. 

Pirie, Susan E B and Kieren, Thomas E (1994) Growth in Mathematical 
Understanding: How can we characterise it and how can we 
represent it? Educational Studies in Mathematics, vol 26, pp 165-190. 

Pozzi, Stefano, Noss, Richard, and Hoyles, Celia (1998) Tools in Practice, 
Mathematics in Use, Educational Studies in Mathematics, vol 36, pp 105-
122. 

Wedege, Tina (2002) ‘Mathematics – That’s what I can’t do’: People’s 
affective and social relationship with mathematics, Literacy and 
Numeracy Studies, vol 11, no 9, pp 63-78. 

 



 F l e x i b l e  M a t h e m a t i c a l  U n d e r s t a n d i n g   
  

 

 
  
60 L I T E R A C Y  &  N U M E R A C Y  S T U D I E S